Dusty Pierson no it doesn't cost anything,and they can find a good home for your dog.
The APL is a non-profit organization dedicated to saving animals from negative outcomes and placing them with forever homes. Go there, tell them your situation and surrender your dog. I've never heard of a shelter charging someone money to rehome a pet.
Sometimes they do charge a little bit as a donation if I were you I would call your local pet shelter and ask
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